We move on to the next part of §3. Application of the Algebraic Signs to Logic.
The Signs of Inclusion, Equality, Etc.
I shall follow Boole in taking the sign of equality to signify identity. Thus, if denotes the Vice-President of the United States, and the President of the Senate of the United States,
means that every Vice-President of the United States is President of the Senate, and every President of the United States Senate is Vice-President.
The sign “less than” is to be so taken that
means that every Frenchman is a man, but there are men besides Frenchmen. Drobisch has used this sign in the same sense. It will follow from these significations of and that the sign (or “as small as”) will mean “is”. Thus,
means “every Frenchman is a man”, without saying whether there are any other men or not. So,
will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. These significations of and plainly conform to the indispensable conditions. Upon the transitive character of these relations the syllogism depends, for by virtue of it, from
|we can infer that|
that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal.
But not only do the significations of and here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.
So, to write is to say that is part of just as to write is to say that Frenchmen are part of men. Indeed, if then the number of Frenchmen is less than the number of men, and if then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.
(Peirce, CP 3.66)
The quantifier mapping from terms to their numbers that Peirce signifies by means of the square bracket notation has one of its principal uses in providing a basis for the computation of frequencies, probabilities, and all the other statistical measures that can be constructed from these, and thus in affording a “principle of correspondence” between probability theory and its limiting case in the forms of logic.
This brings us once again to the relativity of contingency and necessity, as one way of approaching necessity is through the avenue of probability, describing necessity as a probability of 1, but the whole apparatus of probability theory only figures in if it is cast against the backdrop of probability space axioms, the reference class of distributions, and the sample space that we cannot help but abduce on the scene of observations. Aye, there’s the snake eyes. And with them we can see that there is always an irreducible quantum of facticity to all our necessities. More plainly spoken, it takes a fairly complex conceptual infrastructure just to begin speaking of probabilities, and this setting can only be set up by means of abductive, fallible, hypothetical, and inherently risky mental acts.
Pragmatic thinking is the logic of abduction, which is just another way of saying that it addresses the question, What may be hoped? We have to face the possibility that it may be just as impossible to speak of absolute identity with any hope of making practical philosophical sense as it is to speak of absolute simultaneity with any hope of making operational physical sense.